The Kinetic Theory Of Gases

The kinetic theory assumes gas molecules are in random motion, colliding elastically with container walls.
Step 1: Consider a cubic container of side length with molecules, each of mass . Assume 1/3 molecules move along each axis.
Step 2: For a molecule moving with velocity along x-axis, change in momentum upon collision with wall: . Time between collisions: .
Force per molecule: .
Total force on wall: .
Pressure .
Step 3: Since , .
Answer: , where .

The RMS speed of gas molecules is given by: where is the gas constant, is absolute temperature, and is molar mass. Thus, .
Answer: .

Step 1: Formula for mean free path.
The mean free path ( ) is the average distance a molecule travels before colliding with another molecule. It is given by the formula:
Step 2: Analyzing the options.

- (1) : This is incorrect. The formula for the mean free path includes and not just .
- (2) : This is incorrect.
- (3) : Correct. This is the standard formula for the mean free path.
- (4) : This is incorrect because the formula involves .

Step 3: Conclusion.
The correct formula for the mean free path is , corresponding to option (3).

Step 1: Kinetic Theory of Gases.
According to the kinetic theory of gases, gas molecules are in continuous random motion, colliding with the walls of the container. These collisions exert a force on the walls, and the pressure exerted by the gas on the walls is the average force per unit area.
Step 2: Kinetic Energy and Pressure.
The kinetic energy of a single molecule is , where is the mass of the molecule and is its velocity. The rate at which a molecule strikes the walls of the container is proportional to the velocity of the molecules. The pressure is the force per unit area. Using the kinetic theory, we derive the expression for pressure as: where is the number of molecules per unit volume and is the mean square velocity.
Step 3: Conclusion.
Thus, the pressure exerted by gas molecules on the walls of the container is proportional to the number of molecules, the mass of each molecule, and the mean square velocity of the molecules.

The total translational kinetic energy of an ideal gas is given by the formula: At Standard Temperature and Pressure (S.T.P.):
Pressure,
Volume,
Substituting the values: The kinetic energy of 1 litre of an ideal gas at S.T.P. is approximately 152 Joules.